Down with kinds: adding dependent heterogeneous equality to FC (Extended Version)

FC, the core language of the Glasgow Haskell Compiler, is an explicitly-typed variant of System F with first-class type equality proofs (i.e. coercions). This extensible proof system forms the foundation for type system extensions such as type families (typelevel functions) and Generalized Algebraic Datatypes (GADTs). Such features, especially in conjunction with kind polymorphism and datatype promotion, support expressive compile-time reasoning. However, the core language lacks explicit kind equality proofs. As a result, type-level computation does not have access to kindlevel functions or promoted GADTs, the type-level analogues to expression-level features that have been so useful. In this paper, we eliminate such discrepancies by unifying types and kinds. Our approach is based on dependent type systems with heterogeneous equality and the “Type-in-Type” axiom, yet it preserves the metatheoretic properties of FC. In particular, type checking is simple, decidable and syntax directed. We prove the preservation and progress theorems for the extended language.